2020/09/06 by Arkadev Chattopadhyay, Ankit Garg, Chattopadhyay, Arkadev +3
Computer Science · #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Cryptography and Data Security #FOS: Computer and information sciences #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2009.02717
openalex publication_date 2020/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n3) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).